Abstract
The author studies a family of nonlinear integral flows that involve Riesz potentials on Riemannian manifolds. In the Hardy-Littlewood-Sobolev (HLS for short) subcritical regime, he presents a precise blow-up profile exhibited by the flows. In the HLS critical regime, by introducing a dual Q curvature he demonstrates the concentration-compactness phenomenon. If, in addition, the integral kernel matches with the Green’s function of a conformally invariant elliptic operator, this critical flow can be considered as a dual Yamabe flow. Convergence is then established on the unit spheres, which is also valid on certain locally conformally flat manifolds.
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Acknowledgement
The author would like to express his gratitude to Tianling Jin at HKUST for the friendship and fruitful collaboration which has inspired some of the ideas in the current paper.
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Conflicts of interest The authors declare no conflicts of interest.
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In memory of my father
This work was supported by the National Natural Science Foundation of China (Nos. 12325104, 12271028).
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Xiong, J. A Dual Yamabe Flow and Related Integral Flows. Chin. Ann. Math. Ser. B 45, 319–348 (2024). https://doi.org/10.1007/s11401-024-0019-3
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DOI: https://doi.org/10.1007/s11401-024-0019-3